Let $V\subset\mathbb C^n$ (or $\subset\mathbb R^n$) be an algebraic variety corresponding to an ideal generated by polynomials $f_1,\dots,f_k$. Suppose the dimension of $V$ is $n-l$. One cannot always choose $k=l$, so the variety can be overdetermined in the sense that $k>l$. Denote by $f\colon\mathbb C^n\to\mathbb C^k$ the map whose components are the polynomials.
I have understood (see e.g. this MSE question) that a point $p\in V$ is smooth if and only if the Jacobian $Df(p)$ has rank $l$. I am under the impression that a smooth variety is also a smooth submanifold of $\mathbb C^n$ in the sense of differential geometry1, and I want a better geometric understanding of this implication. I struggle to see how this works in the case $k>l$.
If $k=l$, smoothness in the analytic sense is a simple consequence of the implicit function theorem or one can simply take the Jacobian having maximal rank as the definition. When $k>l$, I can apply the implicit function theorem to the $l$ (upon reordering) first polynomials $f_1,\dots,f_l$ and get a smooth $(n-l)$-dimensional submanifold $W$ of $\mathbb C^n$. But why do the additional polynomials $f_{l+1},\dots,f_k$ also vanish on $W$ near $p$?
If I am reading this question correctly, near any point $p$ it is indeed enough to look at $l$ of the polynomials and the variety will be the same. (It seems that $k=l$ means that the variety is a complete intersection, a property that can apparently be made to be the case locally near a smooth point.) The answer to this one makes an essentially identical claim. Thus, my question above can be reformulated: Why is it enough to look at just $l$ of the conditions defining $V$ near a smooth point $p$?
1 I have seen many statements of this claim online, but never with citation. Similarly, Hartshorne states in the first paragraph of I.5 that complex nonsingular varieties are complex manifolds, but doesn't refer to any theorem further in the book.
